Radial basis function methods for optimal control of the convection-diffusion equation

TL;DR

This paper introduces two local radial basis function methods for solving PDE-constrained optimization problems, specifically convection-diffusion control, addressing computational cost and ill-conditioning issues of traditional global RBF approaches.

Contribution

The paper proposes two novel local RBF methods, LAM-DQ and LAM-LAM, combined with a preconditioning technique and extended precision, to improve efficiency and stability in PDE-constrained optimization.

Findings

Local RBF methods outperform global collocation in efficiency.

Preconditioning and extended precision mitigate ill-conditioning.

Methods are highly parallelizable for large-scale problems.

Abstract

PDE-constrained optimization problems have been barely solved by radial basis functions (RBFs) methods [Pearson, 2013]. It is well known that RBF methods can attain an exponential rate of convergence when $C^{\infty}$ kernels are used, also, these techniques, which are truly scattered, are known to be flexible to discretize complex boundaries in several dimensions. On the other hand, exponential convergence implies an exponential growth of the condition number of the Gram matrix associated with these meshfree methods and global collocation techniques are known to be computationally expensive. In this paper, and in the context of optimal constrained optimization problems, we aim to explore a possible answer to both problems. Specifically, we introduce two local RBF methods: LAM-DQ based in the combination of an asymmetric local method (LAM), inspired in local Hermite interpolation (LHI), with the differential quadrature method (DQ), and LAM-LAM which use two times the local asymmetric method. The efficiency of these local methods against global collocation by solving several synthetic convection-diffusion control problems is analyzed. In this article, we also propose a preconditioning technique and treat the ill-conditioning problem by using extended arithmetic precision. We think that these local methods, which are highly parallelizable, shows a possible way to solve massive optimization control problems in an efficient way.